The usual predator-prey model has been successfully applied recently
to the spread of HIV/AIDS in both Canada and the United States. In the
predator-prey ODE model, the rate of increase of the total population
is taken to be equal to the rate at which HIV negative people would
increase without HIV/AIDS minus the rate at which they die. We use this
idea to develop our new model. If we take one year to be the unit of
time, which is the usual interval at which the data on HIV positive
people is published in most countries, then the differential equations
become difference equations. In this paper, we present such a
difference equations model for spread of HIV/ AIDS in any society and
compare the numbers predicted by this model with the actual numbers in
both Canada and the United States. The two sets of numbers are very
close. For appropriate values of the parameters, the model seems to
have a chaotic solution without a chaotic attractor.

A random map is a discrete-time dynamical system in which one of a
number of transformations is randomly selected and applied in each
iteration of the process. In this talk, we study random maps with
position dependent probabilities on the interval. Sufficient conditions
for the existence of absolutely continuous invariant measures for
weakly convex and concave random maps with position dependent
probabilities will be presented.

Linear stability conditions for a first order 3-dimensional discrete
dynamic have been derived in terms of the trace, determinant and sum of
principle minors of the Jacobian evaluated at the equilibrium. The
resulting set of inequalities is an improvement over the Gershgorin
Theorem in that the set of inequalities provides conditions that are
both necessary and sufficient for linear stability in three dimensions
whereas the Gershgorin Theorem provides only sufficient conditions.
Can that same technique be expanded to generalize the linear stability
conditions into higher order discrete dynamics? Can the same method
also best the Gershgorin Theorem in 4-dimensional discrete dynamics?

The probabilities of various biological asymptotic dynamics are
computed for a stable system that is invaded by another competitive
species. The asymptotic behaviors studied include extinction, weak
extinction, permanence, and mutual exclusion. The model used is a
discrete Lotka-Volterra system that model species that compete for the
some resources. Among the results found are that the chance of
permanence occurring in the invaded system is significantly higher than
the probability of permanence in a purely random system, and that
multiple extinctions that include the invading species and one of the
original species are impossible.

Although some preventive vaccines against certain human diseases (such
as measles, rubella, chickenpox etc.) are known to offer lasting
immunity, influenza vaccines offer only a short-term protection which
wanes over time. The vaccine-induced protection against influenza
varies between 70%-90% for young healthy people to approximately
30%-40% in geriatrics or in individuals with a weakened immune
system. The public health objective of this study is to determine,
via mathematical modelling, whether or not such an imperfect vaccine
can be an effective public health tool for controlling (or even
eradicating) the spread of influenza infection within a given
population. A robust deterministic model for influenza transmission,
which incorporates an imperfect vaccine, will be presented and analyzed
qualitatively. The global stability analysis of the model reveals that
the model can exhibit the phenomenon of bi-stability, characterized by
the presence of stable multiple endemic equilibria even when the basic
reproductive number of infection (R0) is less than unity.
The epidemiological consequence of this phenomenon, vis-a-vis the
community-wide control and/or eradication, will be discussed.

The spontaneous formation of order in the form of spatial concentration
patterns in an unstirred chemical medium, supported by dissipation of
chemical free energy, has been considered often since a pioneering
suggestion by Turing and work by Prigogine's group on non-equilibrium
thermodynamics. The best-known experimental example is the oscillatory
Belousov-Zhabotinskii (BZ) reaction, in which target patterns of
outward-moving concentric rings are readily observed. One
widely-studied question is whether "microscopic" fluctuations can
cause nucleate targets, or whether a catalytic, nucleating
heterogeneous center is required. Vidal and Pagola observed spontaneous
activity with no nucleating particles visible at 6-micron resolution;
however Zhang, Forster and Ross argued theoretically that this is
impossible in the steady cycling regime of the BZ reaction. We describe
here an explicit mechanism in a supercritical regime by which
microscopic fluctuations can nucleate activity and reconcile these
results with Zhang et al. Joint work with S.G. Sobel (Hofstra)
and R.J. Field (Univ. of Montana). Partially supported by the NSF.

We investigate the periodic character and boundedness nature of
positive solutions of the difference equation

xn+1=

max

ìí
î

An

xn

,

Bn

xn-1

üý
þ

, n=0,1,¼,

where {An}n=0¥ is a periodic sequence of positive
numbers with period pÎ {1,2,¼} and {Bn}n=0¥ is a periodic sequence of positive numbers with
period qÎ { 1,2,¼}. We show the following:

(i) Each positive solution is either eventually periodic or is
unbounded and nonpersistent.

(ii) For fixed { An}n=0¥ and
{Bn}n=0¥, either every positive solution is
eventually periodic or every positive solution is unbounded and
nonpersistent.

(iii) For fixed { An}n=0¥ and {Bn}n=0¥, if every positive solution is eventually
periodic, then there exists an integer N³ 1 such that every
positive solution is eventually periodic with (not necessarily prime)
period 4Npq.

Fitting experimental data to a chosen mathematical model amounts to
determining the best choice of coefficients or parameters in the
model. In the literature, various methods are used to search the
parameter space. Such problems can often be framed as finding a
contractive map with a particular fixed point. In this talk, we
discuss how the parameters that minimize the "collage distance"
associated with such maps provide in general an excellent starting
point for further optimization.

The most important problem is how to construct convergent difference
scheme. Since the convergence is a consequence of consistency and
stability thus it is necessary to choose those approximating schemes
that are stable. It is naturally to have stability in the norms of the
spaces for which the original problem is stable. For the well-posed
problems of mathematical physics these are the energy spaces where the
squares of the norms express the total energy of the systems. Because
of this, we have to analyze the derivation of the energy estimations in
the differential case and to construct the scheme for which we can
satisfy this derivation in the corresponding Hilbert space in the
discrete case. However, the criteria of consistency and stability
become complicated when applied to the solution of non-linear partial
differential equations. Therefore, the difference scheme has to be
conservative, namely, its conservation laws to be satisfied
identically. Then the non-linearity is not invincible task.

The conservation properties of the mass, momentum, and kinetic energy
equations for incompressible flow are specified as analytical
requirements for a proper set of discrete equations. In present work we
summarize some of the analytical requirements necessary to be satisfied
of the difference scheme. For illustration the vectorial operator
splitting numerical scheme is examined for its conservation properties
and other requirements. It is proven that the difference approximation
of the advection operator conserves square of velocity components and
the kinetic energy like the differential operator does, while pressure
term conserves only the kinetic energy. Therefore, strong stability in
solving higher Reynolds number flows can be achieved, which is
confirmed through various numerical results.

In the theory of analytic dynamical systems we make an extensive use of
normal forms in the neighbourhood of fixed points of diffeomorphisms or
singular points of vector fields. Generically the normalizing changes
of coordinates diverge and the analytic classification of singularities
under changes of coordinates involves functional moduli. An explanation
is given by unfolding the situation and studying the dynamics of the
unfolded system. We present here a complete modulus for a germ of
generic family unfolding a diffeomorphism with a parabolic point and
also for a family unfolding a generic saddle-node of a 2-dimensional
vector field and discuss the geometry of the systems in the family.

In this talk, we will discuss the long-term behavior of solutions of a
class of nonautonomous delay differential equations. By appealing to
the theory of monotone discrete-time dynamical systems, we obtain a
threshold-type result on the global dynamics for scalar periodic delay
differential equations, which is then lifted to a class of
asymptotically periodic delay differential equations. If time permits,
I will give an example for the application of this result to m-species
competitive systems with stage structure.

Consider a predator-prey system with nonmonotonic functional response.
We show that in this case there exists a Bogdanov-Takens bifurcation
point of codimension 3, which acts as an organizing center for the
system. I shall talk about various sequences of bifurcations when the
death rate of the predator is varied. The bifurcation sequences
involving Hopf bifurcations, homoclinic bifurcations as well as the
saddle-node bifurcations of limit cycles are determined using
information from the complete study of the Bogdanov-Takens bifurcation
point of codimension 3 and the geometry of the system.